So, I have noted that sodium has 11 protons, whereas the hydrogen has only 1 proton. So sodium has a heavier nucleus having a charge 11 times greater than that of hydrogen. And sodium has 11 electrons while hydrogen has only 1.

I wanted to argue that the difference of the number of protons and electrons somehow affects the spin-orbit coupling. But in sodium the nucelus with Z=11 is "screened" by 10 negative charges. So we have 1 free electron, exactly like the hydrogen atom! So shouldn't we end up with the same energy diagram?

I am very confused about this. Any explanation would be greatly appreciated.

It's because Sodium starts out with 10 extra electrons that fill up the lowest shells. Therefore these levels aren't available to the valence electron. Cover over the lowest levels of Hydrogen (principal quantum number 1 and 2), and you'll see that Sodium levels match a subset of Hydrogen levels.